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mathematics
precalculus

Questions and Answers of Precalculus

Evaluate the iterated integral by converting to polar coordinates. •1/2 í-y² xv² dx dy /1-y2 /3y
Evaluate the iterated integral by converting to polar coordinates. Va2-y2 -Ja2-y² (2х + у) dx dy
Evaluate the iterated integral by converting to polar coordinates. (V4-x2 ex²-y² dy dx o Jo
(a) A cylindrical drill with radius r1 is used to bore a hole through the center of a sphere of radius r2. Find the volume of the ring-shaped solid that remains.(b) Express the volume in part (a) in
Use polar coordinates to find the volume of the given solid.Inside both the cylinder x2 + y2 = 4 and the ellipsoid 4x2 + 4y2 + z2 = 64
Use polar coordinates to find the volume of the given solid. Bounded by the paraboloids z = 6 - x2 - y2 and z = 2x2 + 2y2
Use polar coordinates to find the volume of the given solid.Above the cone z = √x2 + y2 and below the sphere x2 + y2 + z2 = 1
Use polar coordinates to find the volume of the given solid.Bounded by the paraboloid z = 1 + 2x2 + 2y2 and the plane z = 7 in the first octant
Use polar coordinates to find the volume of the given solid.A sphere of radius a
Use polar coordinates to find the volume of the given solid.Inside the sphere x2 + y2 + z2 = 16 and outside the cylinder x2 + y2 = 4
Use polar coordinates to find the volume of the given solid.Below the plane 2x + y + z = 4 and above the disk x2 + y2 < 1
Use polar coordinates to find the volume of the given solid.Below the cone z = √x2 + y2 and above the ring 1 < x2 + y2 < 4
Use polar coordinates to find the volume of the given solid.Under the paraboloid z = x2 + y2 and above the disk x2 + y2 < 25
Use a double integral to find the area of the region.The region inside the cardioid r = 1 + cos θ and outside the circle r = 3 cos θ
Use a double integral to find the area of the region.The region inside the circle (x - 1)2 + y2 = 1 and outside the circle x2 + y2 = 1
Use a double integral to find the area of the region.One loop of the rose r = cos 3θ
Evaluate the given integral by changing to polar coordinates.where D is the region in the first quadrant that lies between the circles x2 + y2 = 4 and x2 + y2 = 2x I,x dA,
Evaluate the given integral by changing to polar coordinates.where R = {(x, y) | 1 < x2 + y2 < 4, 0 < y < x} Se arctan(y/x) dA, JR
Evaluate the given integral by changing to polar coordinates.where D is the disk with center the origin and radius 2 S, cos /x? + y² dA,
Evaluate the given integral by changing to polar coordinates.where D is the region bounded by the semicircle x = s4 - y2 and the y-axis S,e-° dA, ex²-y2 D
Evaluate the given integral by changing to polar coordinates.where R is the region that lies between the circles x2 + y2 = a2 and x2 + y2 = b2 with 0 < a < b y? dA, ,2 + y? .2
Evaluate the given integral by changing to polar coordinates.where R is the region in the first quadrant between the circles with center the origin and radii 1 and 3 SS, sin(x? + y²) dA,
Evaluate the given integral by changing to polar coordinates.where R is the region in the first quadrant enclosed by the circle x2 + y2 = 4 and the lines x = 0 and y = x y) dA, П. (2х
Evaluate the given integral by changing to polar coordinates.where D is the top half of the disk with center the origin and radius 5 SS,r*y dA,
Sketch the region whose area is given by the integral and evaluate the integral. 2 sine r dr de /2 Jo
Sketch the region whose area is given by the integral and evaluate the integral. СЗп/4 (2 rdr de Уя/4
A region R is shown. Decide whether to use polar coordinates or rectangular coordinates and write as an iterated integral, where f is an arbitrary continuous function on R. le f(x, y) dA ул R
A region R is shown. Decide whether to use polar coordinates or rectangular coordinates and write as an iterated integral, where f is an arbitrary continuous function on R. le f(x, y) dA y -1
A region R is shown. Decide whether to use polar coordinates or rectangular coordinates and write as an iterated integral, where f is an arbitrary continuous function on R. le f(x, y) dA R -1 1
A region R is shown. Decide whether to use polar coordinates or rectangular coordinates and write as an iterated integral, where f is an arbitrary continuous function on R. le f(x, y) dA y R \2
In evaluating a double integral over a region D, a sum of iterated integrals was obtained as follows:Sketch the region D and express the double integral as an iterated integral with reversed order of
Find the averge value of f over the region D.f (x, y) = x sin y, D is enclosed by the curves y = 0, y = x2, and x = 1
Find the averge value of f over the region D.f (x, y) = xy, D is the triangle with vertices (0, 0), (1, 0), and (1, 3)
Use Property 11 to estimate the value of the integral.T is the triangle enclosed by the lines y = 0, y = 2x, and x = 1 || sin*(x + y) dA,
Use Property 11 to estimate the value of the integral. | V4 - x²y² dA, S = {(x, y) | x² + y² < 1, x > 0}
Express D as a union of regions of type I or type II and evaluate the integral. y dA ул УА х3у—уз (1, 1) y= (x+ 1)²/ D -1 -1 х х -1 -1
Express D as a union of regions of type I or type II and evaluate the integral. .2 dA ул УА х3у—уз (1, 1) y= (x+ 1)²/ D -1 -1 х х -1 -1
Evaluate the integral by reversing the order of integration. dx dy e** y 8 (2
Evaluate the integral by reversing the order of integration. *T/2 cos x V1 + cos²x dx dy o Jarcsin y
Evaluate the integral by reversing the order of integration. – 1) dx dy '2 y cos(x³ o Jy/2
Evaluate the integral by reversing the order of integration. Vy3 + 1 dy dx
Evaluate the integral by reversing the order of integration. /y sin y dy dx
Evaluate the integral by reversing the order of integration. Ceахdy Jo ЈЗу dx dy et?
Sketch the region of integration and change the order of integration. C f(x, y) dy dx T/4 arctan x Jo
Sketch the region of integration and change the order of integration. 2 (In x f(x, y) dy dx 1
Sketch the region of integration and change the order of integration. V4-y2 f(x, y) dx dy |-2 Jo
Sketch the region of integration and change the order of integration. °T/2 f(x, y) dy dx cos x
Sketch the region of integration and change the order of integration. f(x, y) dy dx Jx'
Sketch the region of integration and change the order of integration. f(x, y) dx dy
Use a computer algebra system to find the exact volume of the solid.Enclosed by z = x2 + y2 and z = 2y
Use a computer algebra system to find the exact volume of the solid.Enclosed by z = 1 - x2 - y2 and z = 0
Use a computer algebra system to find the exact volume of the solid.Between the paraboloids z = 2x2 + y2 and z = 8 - x2 - 2y2 and inside the cylinder x2 + y2 = 1
Use a computer algebra system to find the exact volume of the solid.Under the surface z = x3y4 + xy2 and above the region bounded by the curves y = x3 - x and y = x2 + x for x > 0
Sketch the solid whose volume is given by the iterated integral. '1–x? (1 – x) dy dx
Sketch the solid whose volume is given by the iterated integral. 1-x (1 – x – y) dy dx
Find the volume of the solid by subtracting two volumes.The solid in the first octant under the plane z = x + y, above the surface z = xy, and enclosed by the surfaces x = 0, y = 0, and x2 + y2 = 4
Find the volume of the solid by subtracting two volumes.The solid under the plane z = 3, above the plane z = y, and between the parabolic cylinders y = x2 and y = 1 - x2
Find the volume of the solid by subtracting two volumes.The solid enclosed by the parabolic cylinder y = x2 and the planes z = 3y, z = 2 + y
Find the volume of the solid by subtracting two volumes.The solid enclosed by the parabolic cylinders y = 1 - x2, y = x2 - 1 and the planes x + y + z = 2, 2x + 2y - z + 10 = 0
Find the approximate volume of the solid in the first octant that is bounded by the planes y = x, z = 0, and z = x and the cylinder y = cos x. (Use a graphing device to estimate the points of
Use a graphing calculator or computer to estimate the x-coordinates of the points of intersection of the curves y = x4 and y = 3x - x2. If D is the region bounded by these curves, estimate dA.
Find the volume of the given solid.Bounded by the cylinders x2 + y2 = r2 and y2 + z2 = r2
Find the volume of the given solid.Bounded by the cylinder x2 + y2 = 1 and the planes y = z, x = 0, z = 0 in the first octant
Find the volume of the given solid.Bounded by the cylinder y2 + z2 = 4 and the planes x = 2y, x = 0, z = 0 in the first octant
Find the volume of the given solid.Enclosed by the cylinders z = x2, y = x2 and the planes z = 0, y = 4
Find the volume of the given solid.Bounded by the planes z = x, y = x, x + y = 2, and z = 0
Find the volume of the given solid.The tetrahedron enclosed by the coordinate planes and the plane 2x + y + z = 4
Find the volume of the given solid.Enclosed by the paraboloid z = x2 + y2 + 1 and the planes x = 0, y = 0, z = 0, and x + y = 2
Find the volume of the given solid.Under the surface z = xy and above the triangle with vertices (1, 1), (4, 1), and (1, 2)
Find the volume of the given solid.Under the surface z = 1 + x2y2 and above the region enclosed by x = y2 and x = 4
Find the volume of the given solid.Under the plane 3x + 2y - z = 0 and above the region enclosed by the parabolas y = x2 and x = y2
Evaluate the double integral.D is the triangular region with vertices (0, 0), (1, 1), and (4, 0) || y dA,
Evaluate the double integral.D is bounded by the circle with center the origin and radius 2 y) dA, (2х D
Evaluate the double integral.D is enclosed by the quarter-circle y = √1 - x2 , x > 0, and the axes ху dA,
Evaluate the double integral.D is the triangular region with vertices (0, 1), (1, 2), (4, 1) y? dA, D.
Evaluate the double integral.D is bounded by y = x, y = x3, x > 0 || (x² + 2y) dA,
Evaluate the double integral.D is bounded by y = 0, y = x2, x = 1 x cos y dA,
Set up iterated integrals for both orders of integration. Then evaluate the double integral using the easier order and explain why it’s easier. I| y?e*y dA, D is bounded by y = x, y = 4, x = 0
Set up iterated integrals for both orders of integration. Then evaluate the double integral using the easier order and explain why it’s easier. y dA, Dis bounded by y = x – 2, x = y?
Express D as a region of type I and also as a region of type II. Then evaluate the double integral in two ways. xy dA, Dis enclosed by the curves y = x², y = 3x
Express D as a region of type I and also as a region of type II. Then evaluate the double integral in two ways. || x dA, D is enclosed by the lines y = x, y = 0, x = 1
Draw an example of a region that is(a) both type I and type II(b) neither type I nor type II
Draw an example of a region that is(a) type I but not type II(b) type II but not type I
Evaluate the double integral. yVx? - y2 dA, D= {(x, y) | 0
Evaluate the double integral. || e-s* dA, D= {(x, y) | 0 < y < 3, 0 < x < y}
Evaluate the double integral. (2x + y) dA, D = {(x, y) | 1 < y < 2, y – 1 < x < 1}
Evaluate the double integral. - dA, D= {(x, y) | 0 < x< 4,0 < y< /x} x? + 1
Evaluate the iterated integral. V1 + e° dw dv
Evaluate the iterated integral. *s2 C cos(s') dt ds Jo Jo
Evaluate the iterated integral. x sin y dy dx T/2 Jo x.
Evaluate the iterated integral. хеdx dy v3
Evaluate the iterated integral. ²y dx dy
Evaluate the iterated integral. '5 х dy dx *5 (8х л 2y)
(a) In what way are the theorems of Fubini and Clairaut similar?(b) If f (x, y) is continuous on [a, b] x [c, d] andfor a < x < b, c < y < d, show that gxy = gyx = f (x, y). g(x, y) = [ L
Use a CAS to compute the iterated integrals
Use symmetry to evaluate the double integral.Do the answers contradict Fubini’s Theorem? Explain what is happening. (1 + x²sin y + y²sin x) dA, R=[-T, ] × [-7, T]
Use symmetry to evaluate the double integral. - dA, 1 + x4 R= {(x, y) | –1
Find the average value of f over the given rectangle.f (x, y) = ey√x + ey , R = [0, 4] x [0, 1]
Find the average value of f over the given rectangle.f (x, y) = x2y, R has vertices (-1, 0), (-1, 5), (1, 5), (1, 0)
Graph the solid that lies between the surfaces z = e2x2 cos(x2 + y2) and z = 2 - x2 - y2 for |x| < 1, |y| < 1. Use a computer algebra system to approximate the volume of this solid correct to

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